The Red Devils earned a 1-1 draw in the first leg of the last-16 clash at the Santiago Bernabeu, but could have carved out a lead had Real goalkeeper Diego Lopez not regularly thwarted the Dutchman.
And, after the game, Van Persie lamented his inability to hand Sir Alex Ferguson's side a potentially decisive victory in the Spanish capital.
"I had three in the second half, especially the other when I hit the bar," the former Arsenal hitman told Sky Sports.
"That was a big chance, I should have put that in. I slightly mis-kicked it, I didn't hit it properly. So with a bit of luck, it could have gone in but it didn't.
"In the dying seconds, their keeper made a really good save because I was counting on that one to go in."
United goalkeeper David de Gea was also worked hard on his return to Spain, and Van Persie admitted either team could have come away with a win.
"Yes, for sure [it could have gone either way]," he added. "They had a couple of good chances and I think if you said it would finish 1-1 before the game, most of us would take it.
"Away from home, it's a good result but for the people who watched the game, they have seen chances on both sides and it was a great game to watch."
The result means that Real has to score when it travels to England, but the 29-year-old doesn't believe that fact counts for much.
When asked whether United has the advantage, he said: "Well slightly. At this level I don't really think these kind of teams care if they play at home or away.
"We don't really care, they don't really care and it is only halftime now. We will be ready for the second half."